Chapter 19: Exercises - Lineup Optimization

Section 19.1: Lineup Net Rating Calculation

Exercise 19.1 (Basic)

A five-man lineup has played 150 possessions together, scoring 168 points and allowing 153 points. Calculate: a) Offensive Rating b) Defensive Rating c) Net Rating d) Interpret whether this lineup is performing above or below league average (assume league average is +0.0)

Exercise 19.2 (Basic)

Given the following lineup statistics: - FGA: 120 - FTA: 40 - OREB: 12 - Turnovers: 18

Calculate the estimated number of possessions using the standard formula.

Exercise 19.3 (Intermediate)

A team's starting lineup has played 800 possessions with a Net Rating of +8.5. The bench unit has played 400 possessions with a Net Rating of -4.2. Calculate the team's overall Net Rating weighted by possessions.

Exercise 19.4 (Intermediate)

Lineup A has a Net Rating of +12.0 based on 100 possessions. Lineup B has a Net Rating of +6.0 based on 400 possessions. a) Calculate the standard error for each lineup's Net Rating b) Which lineup's rating should we trust more and why? c) Construct 95% confidence intervals for both lineups

Exercise 19.5 (Advanced)

Implement a function that calculates luck-adjusted Net Rating by regressing three-point shooting toward league average. The function should take the following inputs: - Team 3PA, Team 3PM - Opponent 3PA, Opponent 3PM - Points scored, Points allowed, Possessions - League average 3P% (default 36%)

Section 19.2: Two-Man and Three-Man Combinations

Exercise 19.6 (Basic)

Player A and Player B have played 500 minutes together with a Net Rating of +7.2. When Player A plays without Player B, the team's Net Rating is +2.1. Calculate the synergy score for this pairing.

Exercise 19.7 (Basic)

Explain why two-man combination analysis provides more reliable insights than five-man lineup analysis despite being less complete.

Exercise 19.8 (Intermediate)

Given a team with 10 players: a) How many unique two-man combinations are possible? b) How many unique three-man combinations are possible? c) How many unique five-man lineups are possible?

Exercise 19.9 (Intermediate)

The following on/off data exists for a three-man core: - All three on: +14.2 Net Rating (300 possessions) - Player A + B only: +8.1 (150 possessions) - Player A + C only: +5.8 (120 possessions) - Player B + C only: +6.5 (130 possessions) - None of the three: -2.8 (200 possessions)

Analyze the synergy effects of this trio and identify which pairings are most dependent on the third player.

Exercise 19.10 (Advanced)

Write Python code that takes a list of player IDs and play-by-play data to identify all three-man combinations with at least 100 minutes played and sort them by Net Rating.

Section 19.3: Five-Man Lineup Evaluation

Exercise 19.11 (Basic)

List Dean Oliver's Four Factors and explain why each matters for lineup evaluation.

Exercise 19.12 (Basic)

A lineup has the following Four Factors: - eFG%: 54.2% - TOV%: 12.1% - ORB%: 28.5% - FT Rate: 0.32

Compare each metric to typical NBA averages and identify the lineup's strengths and weaknesses.

Exercise 19.13 (Intermediate)

Classify the following lineup into an archetype (closing, transition, defensive, offensive, development, or rest): - Average minutes per game of starters: 32, 31, 29, 18, 15 - Pace: 108 possessions per 48 - Offensive Rating: 118.2 - Defensive Rating: 112.5 - Average player experience: 6.2 years - Clutch minutes: 45

Exercise 19.14 (Intermediate)

Why might a lineup with a +15 Net Rating in 80 possessions be less valuable than a lineup with a +5 Net Rating in 500 possessions? Discuss both statistical and practical considerations.

Exercise 19.15 (Advanced)

Design a lineup evaluation framework that weights the Four Factors based on their correlation with winning. Research or estimate the appropriate weights and justify your choices.

Section 19.4: Rotation Analysis and Patterns

Exercise 19.16 (Basic)

In a typical NBA rotation: a) How long do starters usually play in their first stint? b) When do the first substitutions typically occur? c) What is the average stint length for a rotation player?

Exercise 19.17 (Basic)

A player enters games at the 6-minute mark of the first quarter on average and exits at the 9-minute mark of the second quarter. Calculate their typical first-half stint length.

Exercise 19.18 (Intermediate)

Research suggests that player performance degrades after 10-12 continuous minutes. If a team wants to play their star 36 minutes per game while never exceeding 12-minute stints: a) What is the minimum number of stints needed? b) Design a rotation pattern that accomplishes this goal c) What are the tradeoffs of this approach?

Exercise 19.19 (Intermediate)

Analyze the following substitution pattern for a 48-minute game: - Starter A: Minutes 0-8, 12-20, 24-32, 36-48 (total: 36 min) - Starter B: Minutes 0-7, 10-19, 24-31, 34-48 (total: 37 min)

Calculate the overlap minutes between these two players and identify potential issues with this rotation.

Exercise 19.20 (Advanced)

Create a visualization function that takes player minute-by-minute data and produces a rotation chart showing when each player was on the court throughout a game.

Section 19.5: Lineup Construction Principles

Exercise 19.21 (Basic)

What are the five essential skills to balance in lineup construction? Give an example of a player archetype that provides each skill.

Exercise 19.22 (Basic)

A lineup has the following three-point shooting distribution: - Player 1: 42% on 5.5 attempts/game - Player 2: 38% on 4.2 attempts/game - Player 3: 31% on 1.8 attempts/game - Player 4: 34% on 3.1 attempts/game - Player 5: 25% on 0.5 attempts/game

Calculate the lineup's expected three-point percentage and number of reliable shooters (>35% on 2+ attempts).

Exercise 19.23 (Intermediate)

Calculate a defensive versatility score for a lineup where: - Player 1 can guard positions 1-2 - Player 2 can guard positions 1-3 - Player 3 can guard positions 2-4 - Player 4 can guard positions 3-5 - Player 5 can guard positions 4-5

Is there any position that lacks adequate coverage?

Exercise 19.24 (Intermediate)

Design two different five-man lineups for a team that has the following player types available: - 2 traditional centers (rim protection, limited shooting) - 2 stretch fours (shooting, some rim protection) - 3 wings (shooting, perimeter defense) - 2 combo guards (ball handling, shooting) - 1 traditional point guard (ball handling, average shooting)

Create one "pace and space" lineup and one "traditional" lineup, explaining your choices.

Exercise 19.25 (Advanced)

Implement a lineup spacing calculator that takes player shooting statistics and outputs a spacing score from 0-100 based on the number of reliable shooters and expected three-point percentage.

Section 19.6: Stagger Principles

Exercise 19.26 (Basic)

Explain the concept of "staggering" star players and why it is strategically important.

Exercise 19.27 (Basic)

A team has two stars who both need to play 35 minutes per game. If they want at least one star on the court at all times: a) What is the minimum overlap they must have? b) What is the maximum minutes they can stagger?

Exercise 19.28 (Intermediate)

Create a 48-minute schedule for two stars (Player A: 36 minutes, Player B: 32 minutes) that: - Ensures at least one star is on court for at least 44 minutes - Has both stars on court for closing minutes (final 6 minutes) - Provides each player at least 8 minutes of rest in two stints

Exercise 19.29 (Intermediate)

Analyze the tradeoffs between: a) Staggering two ball-dominant guards b) Playing two ball-dominant guards together Consider both offensive and defensive implications.

Exercise 19.30 (Advanced)

Implement a stagger optimization algorithm that takes target minutes for multiple star players and outputs a minute-by-minute schedule that maximizes star coverage while respecting rest constraints.

Section 19.7: Closing Lineup Optimization

Exercise 19.31 (Basic)

Define "clutch time" according to the NBA's official definition and explain why closing lineups are disproportionately important.

Exercise 19.32 (Basic)

List five skills that are especially valuable for closing lineups and explain why each matters in late-game situations.

Exercise 19.33 (Intermediate)

Given the following player data, rank them for inclusion in a closing lineup when protecting a 3-point lead with 2 minutes remaining:

Exercise 19.34 (Intermediate)

How should a closing lineup differ when: a) Protecting a lead b) Chasing a deficit c) In a tie game Provide specific lineup construction recommendations for each scenario.

Exercise 19.35 (Advanced)

Build a closing lineup evaluation function that scores potential lineups based on weighted factors (free throw shooting, ball security, shot creation, defensive versatility, rebounding) and returns the top 3 lineup options from a roster.

Section 19.8: Sample Size Challenges

Exercise 19.36 (Basic)

Why is the standard error of Net Rating approximately 11 points per 100 possessions / sqrt(n)?

Exercise 19.37 (Basic)

A lineup has played 200 possessions with a Net Rating of +8.0. Calculate the 95% confidence interval for this lineup's true talent level.

Exercise 19.38 (Intermediate)

Given that Net Rating stabilizes after approximately 1000 possessions: a) How many minutes does this represent (approximately)? b) How many games would a starting lineup need to play together to reach this threshold? c) Why do most lineups never reach this threshold?

Exercise 19.39 (Intermediate)

Explain why Bayesian approaches are particularly useful for lineup analysis. What should the prior distribution represent, and how should it change as sample size increases?

Exercise 19.40 (Advanced)

Implement a Bayesian lineup estimator that takes observed Net Rating and possessions, applies a prior (mean 0, SD 5), and returns the posterior mean, standard deviation, and credible interval.

Section 19.9: Optimization Algorithms

Exercise 19.41 (Basic)

Describe the lineup optimization problem in terms of: a) The objective function b) Key constraints c) Why this is computationally challenging

Exercise 19.42 (Intermediate)

A team has 10 available players with the following maximum minutes: - Players 1-3: 36 minutes each - Players 4-6: 28 minutes each - Players 7-10: 18 minutes each

Calculate the total available player-minutes and verify there are enough to cover 240 player-minutes (5 players x 48 minutes).

Exercise 19.43 (Intermediate)

Compare greedy heuristic approaches to integer linear programming for lineup optimization. What are the advantages and disadvantages of each?

Exercise 19.44 (Advanced)

Design a Monte Carlo simulation that evaluates a rotation strategy by simulating 1000 games and tracking: - Win probability - Star player fatigue - Total expected point differential

Exercise 19.45 (Advanced)

Implement a greedy rotation optimizer that, at each minute of a game, selects the best available lineup while respecting minutes and rest constraints.

Applied Exercises

Exercise 19.46 (Project)

Using NBA lineup data (from Basketball-Reference or NBA Stats API): a) Identify the top 10 two-man combinations by Net Rating (minimum 300 minutes) b) Identify the top 10 three-man combinations by Net Rating (minimum 200 minutes) c) Calculate correlation between two-man and three-man ratings for the same player pairs d) Discuss what these relationships reveal about player synergies

Exercise 19.47 (Project)

Analyze a championship team's rotation patterns: a) Chart the typical rotation through a game b) Calculate stagger rates for their top 3 players c) Identify their most-used closing lineup d) Compare regular season and playoff rotation patterns

Exercise 19.48 (Project)

Build a complete lineup optimization tool that: a) Takes a roster with player ratings and minute constraints b) Calculates all possible five-man lineup combinations c) Applies Bayesian regularization to lineup ratings d) Recommends optimal rotations for different game situations

Exercise 19.49 (Case Study)

Analyze the Golden State Warriors' "Death Lineup" (Curry, Thompson, Iguodala, Green, Barnes/Durant): a) Calculate the lineup's historical Net Rating b) Identify what made this lineup special analytically c) Discuss why it was deployed as a closing lineup d) Analyze the defensive tradeoffs of playing small

Exercise 19.50 (Research)

Investigate whether lineup Net Rating in the regular season predicts playoff success: a) Gather data on team's best lineups from regular season b) Compare to their playoff performance c) Analyze which lineup characteristics translate best to playoffs d) Discuss implications for roster construction

Challenge Problems

Challenge 19.1

Prove mathematically that the standard error of Net Rating is proportional to 1/sqrt(possessions) under the assumption that individual possession outcomes are independent with approximately equal variance.

Challenge 19.2

Design an optimization algorithm that finds the rotation maximizing expected wins over a season while: - Managing player fatigue over 82 games - Accounting for back-to-back games - Adjusting for opponent quality - Respecting player injury risk models

Challenge 19.3

Develop a neural network architecture that predicts lineup performance from player embeddings. Explain how this approach addresses sample size limitations compared to pure empirical lineup ratings.

Challenge 19.4

Create a game theory model for rotation decisions where: - Both teams are optimizing their lineups - Substitution decisions affect opponent reactions - The goal is to find Nash equilibrium rotation strategies

Challenge 19.5

Build a complete lineup analytics dashboard that includes: - Interactive lineup performance comparison - Real-time rotation optimization - Stagger analysis visualization - Sample size warnings and Bayesian adjustments - Historical comparison tools